Normal Distribution

A Normal Distribution is:

also known as a Gaussian distribution and Bell Curve
applied to real-valued random variables
a type of probability distribution
characterized by its mean and standard deviation
applied to many natural phenomena

Normal Distribution Plot

The plot below illustrates a normal distribution where each band has a width of 1 Standard Deviation:

Normal Distribution Probability Density Function

The general form of the normal distribution probability density function is:

$\\f\left ( x \right ) = \frac{1}{\sigma \sqrt{2\pi }}e^{-\frac{1}{2}\left (\frac{x-\mu }{\sigma } \right )^{2}} \\\\\mathrm{\textup{where:}} \\\\f\,\,\,\,\,\mathrm{\textup{function for normal distribution}} \\x\,\,\,\,\,\mathrm{\textup{random variable}} \\\sigma\,\,\,\,\,\mathrm{\textup{standard deviation}} \\\mu \,\,\,\,\,\mathrm{\textup{mean}} \\e\,\,\,\,\,\,\mathrm{\textup{euler's number}} \\\pi \,\,\,\,\,\mathrm{\textup{pi}}$

Mean and Variance

Visually, you can see the function curve shift as the variance (standard deviation squared) and mean change:

Occurrence in Nature

Normal Distributions of randomly selected data points are common in nature. Examples include human:

Blood Pressure
Height
IQ
Weight

Normal Distribution

Normal Distribution Plot

Normal Distribution Probability Density Function

Mean and Variance

Occurrence in Nature

References